Maximization Of Fundamental Frequency Research Articles

A cutting plane approach to maximization of fundamental frequency in truss topology optimization

A cutting plane approach to maximization of fundamental frequency in truss topology optimization

Optimal material and geometry of 2D-FGM tapered microbeams under dynamic and static constraints

The present work applies meta-heuristic multi-objective particle swarm optimization (MOPSO) and non-dominated sorting genetic algorithm (NSGA)-II algorithms to predict optimal shape and material distribution of two-directional functionally graded tapered (2DFG-T) microbeams. The cross-section of the microbeam changes linearly along its length in both height and width directions. The rates of these changes are taken as the geometric design variables. Rather than the simple power law, two different trigonometric functions are introduced to provide various effective and flexible 2D- distributions of metal and ceramic phases. Using Hamilton’s principle, the mathematical model is derived based on the third order shear deformation theory (TSDT) and modified couple stress theory (MCST). Developing a computationally efficient and time saving semi-analytical approach, the governing set of variable-coefficients partial differential equations is solved for the frequency and critical buckling under different boundary conditions. Several constrained single- and multi-objective optimization problems are formulated and solved to determine optimal material and geometry parameters of 2DFG-T microbeams to achieve maximization of fundamental frequency and/or critical buckling load while minimizing the structural mass. Various optimization results demonstrate that including shape parameters in addition to material distribution parameters as design variables results in more economical and flexible profiles that is critical for the manufacture of 2DFG-T microbeams. The following model can be exploited in the optimum design of FG microscale nanobeams including their shapes and material distribution.

Reliability-based topology optimization for fundamental frequency maximization with frequency band constraints

Resonance can be avoided by preventing the structural natural frequencies from falling within the operating frequency range. However, structural natural frequencies are influenced by uncertain parameters, such as material properties and geometric sizes. Therefore, it is essential to consider these uncertainties during the conceptual design phase. To address this problem, a novel reliability-based eigenvalue topology optimization model under frequency-band constraints is established, in which the uncertainties of the material properties and structural size are considered. Owing to the high computational costs of dealing with frequency-band reliability constraints, a frequency-band constraint shifting method (FBCSM) is proposed to ease the heavy computational burden without sacrificing accuracy. In addition, the sensitivities of simple and multiple eigenfrequencies with respect to the design and random variables are derived. The “bound formulation” and “robust formulation” are used to reduce the gray elements and checkerboard phenomenon. Four examples are used to determine the effectiveness of the proposed reliability-based eigenvalue topology model and FBCSM. The results confirmed their superiority in terms of validity and efficiency.

Free vibration analysis and multi-objective optimization of lattice sandwich beams

The free vibration dynamic modellings of sandwich beams with periodic lattice truss cores are established by combining the Bernoulli–Euler beam theory and Timoshenko beam theory, and the fundamental frequency and unit cell mass of lattice sandwich beams are optimized based on the non-dominated sorting genetic algorithm-II (NSGA-II). Firstly, the vibration control equation of lattice sandwich beams is derived using Hamilton’s principle. Then, it is compared and verified by modal experiment and finite element numerical simulation. On this basis, the influence of different truss cores on the structure’s natural frequency is further analyzed. Finally, taking the sheet thickness, core thickness, and radius of the truss as the optimization design variables and taking the structural fundamental frequency maximization and mass minimization as the optimization objectives, the Pareto front is obtained by NSGA-II, and the optimization results are verified by theoretical and numerically. The research results show that the theoretical method in this article can well predict the natural frequencies of sandwich beams with different truss cores. The optimized sandwich structure has higher fundamental frequency with lighter weight, which effectively improves the anti-vibration ability.

Analysis and optimization for vibration of laminated rectangular plates with blended layers

An accurate and straightforward analytical method is proposed for the free vibration of a laminated composite rectangular plate with blended layers. The plate with blended layers is defined here as a laminated plate where some of the outer layers are composed of multiple subareas with different orientation angles of straight fibers. A finite element code is developed as well to verify accuracy of the proposed method. In numerical comparison, the natural frequencies calculated by the present method agree well with those from the FEM code for rectangular plates with two, three, four and five subareas. Subsequently, the use of this analytical method is tested in vibration optimization, i.e. design for maximization of fundamental frequencies. This frequency optimization is formulated by changing the fiber orientation angles in multiple subareas simultaneously, and is carried out by using permutation in simple cases and by Genetic Algorithm (GA) for a larger number of frequency calculations. It is observed that introduction of blended layers is very effective approach to design the dynamic characteristics of laminated composite plates.

Fundamental frequency maximization of composite rectangular plates by sequential permutation search algorithm

Due to the classical thin plate theory based governing equations of composite rectangular plates for dynamic problems are linear homogeneous equations on bending stiffness coefficients, a general sequential permutation search (SPS) algorithm is presented to optimize stacking sequences of composite laminates by exploiting convexity of lamination parameters, sensitivity of bending stiffness, and bending-twisting coupling stiffness feature. Numerical examples are conducted for 8-layer composite rectangular plates with different boundaries to maximize the fundamental frequency evaluated by the Rayleigh–Ritz method, and the results are compared with those of layerwise optimization approach and genetic algorithm, demonstrating robustness and efficiency of SPS.

FUNDAMENTAL FREQUENCY MAXIMIZATION DESIGN FOR CONTINUOUS FIBER-REINFORCED COMPOSITE STRUCTURES

Compared with traditional metal-materials, fiber-reinforced composite materials have better performance in many aspects such as strength, stiffness, and fracture resistance. At present, fiber-reinforced composite materials have been widely used in automotive, aerospace, and other industrial fields. This paper proposes a topology optimization method for solving the fundamental frequency maximization of undamped free vibration of continuous fiber-reinforced composite structures. To achieve the simultaneous optimization of the structural topological configuration and the fiber angle. A dynamic topological optimization model is established with the permitted material usage as the constraint and the structure's first-order eigenvalue as the objective function. The model includes density design variables that characterizes the topological configuration of the structure and angular design variables that characterizes the fiber orientation. The analytical sensitivity formulas of the objective function of eigenvalue with respect to density design variables and angle design variables are derived in detail, and the method of moving asymptotes (MMA) is used to solve the optimization problem. Finally, three numerical examples are performed to verify the effectiveness of the proposed method, which includes a static optimization example with the stiffness maximization as the goal and two dynamic optimization examples with the first-order eigenvalue maximization as the goal. The results show that the proposed method can achieve a stable iterative history and fast convergence, and can effectively improve the structural frequency while achieving the integrated optimization of the structural topological configuration and the fiber angle.

Multi-objective optimal design of hybrid composite laminates for minimum cost and maximum fundamental frequency and frequency gaps

The design of hybrid composite laminates made of high-stiffness skin and low-stiffness core layers is presented. By considering vibration characteristics, the objective is the simultaneous maximization of fundamental frequency (or the gap between two consecutive frequencies) and minimization of cost by seeking the optimal stacking sequences of both skin and core layers. By introducing the concept of ground structure in laminate design, an initial stacking sequence consisting of high-stiffness skin and low-stiffness core layers is firstly given, and the design problem is then formulated with mixed discrete and continuous variables by defining a weighted min-max objective function and determining the minimum, where discrete variables represent the existence of each ply in the initial stacking sequence, and continuous variables are used for ply thicknesses. The problem is made explicit with branched multipoint approximate functions, and genetic algorithm (GA) is adopted to optimize discrete variables so that necessary/unnecessary layers from the initial stacking sequence are retained/suppressed. For fitness calculation in GA, a second-level approximation is built with the convex linearization to optimize continuous ply thicknesses of the retained layers. Using that approach, optimal stacking sequences are found for hybrid graphite/epoxy-glass/epoxy laminated plates with different aspect ratios, by considering free vibration.

A consistent methodology for optimal shape design of graphene sheets to maximize their fundamental frequencies considering topological defects

In recent years, shape design of graphene sheets (GSs) by introducing topological defects for enhancing their mechanical behaviors has attracted the attention of scholars. In the present work, we propose a consistent methodology for optimal shape design of GSs using a combination of the molecular mechanics (MM) method, the non-parametric shape optimization method, the phase field crystal (PFC) method, Voronoi tessellation, and molecular dynamics (MD) simulation to maximize their fundamental frequencies. At first, we model GSs as continuum frame models using a link between the MM method and continuum mechanics. Then, we carry out optimal shape design of GSs in fundamental frequency maximization problem based on a developed shape optimization method for frames. However, the obtained optimal shapes of GSs only consisting of hexagonal carbon rings are unstable that do not satisfy the principle of least action, so we relocate carbon atoms on the optimal shapes by introducing topological defects using the PFC method and Voronoi tessellation. At last, we perform the structural relaxation through MD simulation to determine the final optimal shapes of GSs. We design two examples of GSs and the optimal results show that the fundamental frequencies of GSs can be significantly enhanced according to the optimal shape design methodology.

Maximization of fundamental frequency and buckling load for the optimal stacking sequence design of laminated composite structures

The paper illustrates the application of a two-level approximation method to the lay-up design of laminated structures for maximization of fundamental frequency and buckling load with design constraints. Previously developed for the mass-minimization design, the approximation method was achieved by starting from an initial design of stacking sequence. Benchmark examples have verified its efficacy in dealing with the mass-reduction problems, but it does not have the capability to address the objective-maximization problems, which is mainly due to the limitation of the second-level approximate problem. In this work, this method is improved and extended for the consideration of objective maximization with more design constraints. The second-level approximate problem is reconstructed with mixed direct/reciprocal design variables, suitable for solving maximization problems. By varying different initial designs of stacking sequence and conducting repeated runs in the numerical examples, its efficiency is significantly shown after making comparisons with other methods.

Maximization of fundamental frequency of layered composites using differential evolution optimization

Differential evolution optimization is used to find staking sequences for maximization of the natural frequency of symmetric and asymmetric laminates. Clamped and simply supported boundary condition are considered. A meshless numerical method is used to compute free vibrations of square and rectangular plates. Obtained staking sequences for symmetric laminates are in accordance with the literature. Optimized staking sequences for 8 ply asymmetric laminates are also presented. Asymmetric staking sequences produced higher first natural frequencies. Posterior analysis shows that optimized asymmetric staking sequences are hygrothermally stable.

Topology design for maximization of fundamental frequency of couple-stress continuum

In this paper, the topology optimization formulation of couple-stress continuum is investigated for maximizing the fundamental frequency. A modified bound formulation is used to prevent the order s...

Maximization of fundamental frequencies of axially compressed laminated truncated conical shells against fiber orientation

Free vibration analyses of laminated truncated conical shells subjected to axial compressive forces are carried out by employing the Abaqus finite element program. The fundamental frequencies of these truncated conical shells with a given material system are then maximized with respect to fiber orientations by using the golden section method. Through parametric studies, the influences of the end condition, shell length, shell radius ratio and the compressive force on the maximum fundamental frequencies, the associated optimal fiber orientations and the associated vibration modes are demonstrated and discussed.

Optimization of fiber distribution in fiber reinforced composite by using NURBS functions

This research deals with the optimization of short fibers distribution in continuum structures made of Fiber Reinforced Composite (FRC) by adopting an efficient gradient based optimization approach. Motivated by lack of non-heuristic and mesh independent optimization algorithms to obtain the optimum distribution of short fibers through a design domain, Non-Uniform Rational B-spline (NURBS) basis functions have been implemented to define continuous and smooth mesh independent fiber distribution function as well as domain discretization. Thanks to higher order (here quadratic) NURBS basis functions along with their compact support, a drastic reduction in computational time has been obtained by increasing mesh size while the accuracy of the model is maintained. Moreover combination of NURBS with sensitivity based optimization method allows a fast convergence to optimum fiber distribution layout. Minimization of elastic strain energy and maximization of fundamental frequency have been considered as objective functions for static and free vibration problems, respectively; to get the maximum fiber exploitation in the structural element. Nodal volume fraction of fiber was defined as the optimization design variable while a homogenization approach based on the random orientation of short fibers in the matrix has been adopted. Some numerical examples related to the structural response under static loading as well as the free vibration behavior are finally conducted to demonstrate the capability and reliability of the model.

Method for optimizing manipulator’s geometrical parameters and selecting reducers

A geometrical parameters optimization and reducers selection method was proposed for robotic manipulators design. The Lagrangian approach was employed in deriving the dynamic model of a two-DOF manipulator. The flexibility of links and joints was taken into account in the mechanical structure dimensions optimization and reducers selection, in which Timoshenko model was used to discretize the hollow links. Two criteria, i.e. maximization of fundamental frequency and minimization of self-mass/load ratio, were utilized to optimize the manipulators. The NSGA-II (fast elitist nondominated sorting genetic algorithms) was employed to solve the multi-objective optimization problem. How the joints flexibility affects the manipulators design was analyzed and shown in the numerical analysis example. The results indicate that simultaneous consideration of the joints and the links flexibility is very necessary for manipulators optimal design. Finally, several optimal combinations were provided. The effectiveness of the optimization method was proved by comparing with ADAMS simulation results. The self-mass/load ratio error of the two methods is within 10%. The maximum error of the natural frequency by the two methods is 23.74%. The method proposed in this work provides a fast and effective pathway for manipulator design and reducers selection.

An ant colony algorithm applied to lay-up optimization of laminated composite plates

ANT COLONY OPTIMIZATION (ACO) IS A CLASS OF HEURISTIC ALGORITHMS PROPOSED TO SOLVE OPTIMIZATION PROBLEMS. THE IDEA WAS INSPIRED BY THE BEHAVIOR OF REAL ANTS, RELATED TO THEIR ABILITY TO FIND THE SHORTEST PATH BETWEEN THE NEST AND THE FOOD SOURCE. ACO HAS BEEN APPLIED SUCCESSFULLY TO DIFFERENT KINDS OF PROBLEMS. SO, THIS MANUSCRIPT DESCRIBES THE DEVELOPMENT AND APPLICATION OF AN ACO ALGORITHM TO FIND THE OPTIMAL STACKING SEQUENCE OF LAMINATED COMPOSITE PLATES. THE DEVELOPED ACO ALGORITHM IS EVALUATED ON FOUR EXAMPLES FOR SYMMETRIC AND BALANCED LAY-UP. THE RESULTS OF THE FIRST THREE CASES, IN WHICH THE CLASSICAL LAMINATION THEORY IS USED TO OBTAIN THE STRUCTURAL RESPONSE OF RECTANGULAR PLATES, ARE COMPARED TO THOSE OBTAINED FROM THE LITERATURE USING GENETIC ALGORITHMS (GA) AND OTHER ACO ALGORITHM. THE FOURTH EXAMPLE INVESTIGATES THE MAXIMIZATION OF FUNDAMENTAL FREQUENCIES OF RECTANGULAR PLATES WITH CENTRAL HOLES, WHERE THE STRUCTURAL RESPONSE IS OBTAINED BY FINITE ELEMENT ANALYSIS, SHOWING THAT THIS OPTIMIZATION TECHNIQUE MAY BE SUCCESSFULLY APPLIED TO A BROAD CLASS OF STACKING SEQUENCE PROBLEMS FOR LAMINATED COMPOSITES.

Maximization of Fundamental Frequency of Laminated Composite Cylindrical Shells by Ant Colony Algorithm

http://dx.doi.org/10.5028/jatm.v5i1.233The success in developing modern aerospace systems addresses competitive subjects as optimization, reduced costs, sustainability, environment, weight, and safety. There is an increased demand for lighter materials such as laminated composites. In order to match the demand of aeronautical companies, the shell structures are very important. The dynamic behavior of composite structures is also essential to improve the potential applications of these materials. The knowledge of the dynamic response of the cylindrical shell structures is an important issue in their design, in which the ply thicknesses are often preestablished and the ply orientations are usually restricted to a small set of angles due to manufacturing constraints. Obtaining the best stacking sequence of laminated shell may lead to a problem of combinatorial optimization. As this problem is hard to be solved, several techniques have been developed. Ant colony optimization is a class of heuristic optimization algorithms inspired by the behavior of real ants, related to their ability to find the shortest path between the nest and the food source. Thus, this paper deals with optimal stacking sequence of laminated cylindrical composite shells with the aim of maximizing their fundamental frequency using this approach. The ant colony algorithm was implemented in Matlab platform and was linked to Abaqus academic version to compute the structural response. Cylindrical shells with and without a cutout geometry were studied. Fundamental frequencies were maximized for both cases, and results were presented and discussed.

Maximization of fundamental frequency of axially compressed laminated curved panels with cutouts

Free vibration analyses of laminated curved panels with central circular cutouts and subjected to axial compressive forces are carried out by employing the Abaqus finite element program. The fundamental frequencies of these composite laminated curved panels with a given material system are then maximized with respect to fiber orientations by using the golden section method. Through parametric studies, the significant influences of the panel aspect ratio, the panel curvature, the cutout size and the compressive force on the maximum fundamental frequencies, the optimal fiber orientations and the associated fundamental vibration modes of these panels are demonstrated and discussed.

Constrained thickness optimization of rectangular orthotropic fiber-reinforced plate for fundamental frequency maximization

This paper addresses the thickness optimization of rectangular orthotropic fiber-reinforced composite plates by using mathematical programming theory of nonlinear constrained optimization. We optimize the thickness distribution of various composite plates for reaching a maximum in the plate natural frequency, subject to an equality constraint in a plate volume and a number of inequality constraints on the lower and upper bounds of allowable range of thicknesses. For the modeling part of our optimization algorithm, we formulate the kinematical hypothesis of the classical plate theory (CPT), reduced plane stress theory of linearized elasticity, and Hamilton’s principle. For the solution part, we implement the Rayleigh-Ritz semi-analytical solution technique to spatially discretize the weak form of the plate partial differential equation (PDE), and transforming it to a generalized algebraic eigen-problem. For the design optimization part, we use a conjugate Rosen’s gradient projection method. The design variables of the optimization design space are thicknesses of an arbitrary but rectangular partition of the plate into constant-thickness subdivisions. After simulating our algorithm, we noticed many interesting and impressive results, and tried to justify them. The most impressive finding is that, as the fiber orientation of the composite plate rotates, the plate thickness distribution rotates also in the same way, but with different manifestations for different boundary conditions. For parametric studies, we analyze the sensitivity of the algorithm against parameters, like, the degree of fineness in the number of plate partitions into constant-thickness sub-plates, the degree of plate anisotropy, the fiber orientation, the rectangular plate aspect ratio, the upper to lower bounds of allowable thickness ratio, and boundary conditions.

Maximization of Fundamental Frequencies of Axially Compressed Laminated Curved Panels against Fiber Orientation

Maximization of Fundamental Frequencies of Axially Compressed Laminated Curved Panels against Fiber Orientation